An important effect on the critical-current density Jc in a superconducting tape is found when the tape is above a soft ferromagnetic substrate, as is the case with many coated conductors. Here, a model is presented to analyze the complex magnetic interaction between the induced currents in the superconductor and the poles in the magnet. We find that transport Jc decreases at nonzero applied fields when there is a magnetic substrate and this reduction can be controlled by changing the magnet thickness. The model presented here can, therefore, become a useful tool for understanding and optimizing superconductor-magnet systems. © 2010 American Institute of Physics.